Abstract
A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y ∈ G such that for any x ∈ G the nth commutator [x,y, . . . ,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group.
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References
Amayo, R.K., Stewart, I.: Infinite-dimensional Lie Algebras, Noordhoff, Leyden, 1974
Baer, R.: Engelsche Elemente Noetherscher Gruppen. Math. Ann. 133, 256–270 (1957)
Bahturin, Yu.: Identical Relations in Lie Algebras, VNU Science Press, Utrecht, 1987
Bandman, T., Grunewald, F., Greuel, G.-M., Kunyavskii, B., Pfister, G., Plotkin, E.: Two-variable identities for finite solvable groups. C.R. Acad. Sci. Paris, Ser. I 337, 581–586 (2003)
Bandman, T., Greuel, G.-M., Grunewald, F., Kunyavskii, B., Pfister, G., Plotkin, E.: Identities for finite solvable groups and equations in finite simple groups. Compositio Math. to appear
Boffa, M., Point, F.: Identités de Thue–Morse dans les groupes. C.R. Acad. Sci. Paris, Sér. I Math. 312, 667–670 (1991)
Bourbaki, N.: Groupes et algèbres de Lie, Ch. I, Hermann, Paris, 1971
Brandl, R., Wilson, J. S.: Characterization of finite soluble groups by laws in a small number of variables. J. Algebra 116, 334–341 (1988)
Bray, J.N., Wilson, J.S., Wilson, R.A.: A characterization of finite soluble groups by laws in two variables. Bull. London Math. Soc. 37, 179–186 (2005)
Cossey, J., Oates Macdonald, S., Street, A.P.: On the laws of certain finite groups. J. Austral. Math. Soc. 11, 441–489 (1970)
Grunewald, F., Kunyavskii, B., Nikolova, D., Plotkin, E.: Two-variable identities in groups and Lie algebras. Zap. Nauch. Semin. POMI 272, 161–176 (2000); J. Math. Sci. (New York) 116, 2972–2981 (2003)
Huppert, B.: Endliche Gruppen, I, Springer-Verlag, Berlin–Heidelberg–New York, 1979
Huppert, B., Blackburn, N.: Finite Groups, III, Springer-Verlag, Berlin–Heidelberg–New York, 1982
Jacobson, N.: Lie Algebras, Interscience Publ, New York–London, 1962
Koshlukov, P.: Weak polynomial identities for the matrix algebra of order two. J. Algebra 188, 610–625 (1997)
Lang, S.: Algebra, 3d ed., Addison-Wesley, Reading, MA, 1993
Mal'cev, Ju.N., Kuz'min, E.N.: A basis for identities of the algebra of second-order matrices over a finite field. Algebra i Logika 17 (1), 28–32 (1978) (Russian)
Platonov, V.P.: Engel elements and radical in PI-algebras and topological groups. Dokl. Akad. Nauk SSSR 161, 288–291 (1965) (Russian)
Plotkin, B.I.: Radical groups. Mat. Sb. N.S. 37 (79), 507–526 (1955); English transl. in Amer. Math. Soc. Transl. 17 (2), 9–28 (1961)
Plotkin, B.I.: Radical and nil elements in groups. Izv. Vysš. Učebn. Zaved. Matematika (1958), n°1 (2), 130–135 (Russian)
Plotkin, B.I.: Notes on Engel groups and Engel elements in groups. Some generalizations, Izvestija Ural'skogo Universiteta, Ser. Matematika–Mehanika 36 (7), 153–166. (2005) available at http://arXiv.org/math.GR/0406100
Procesi, C.: The Burnside problem. J. Algebra 4, 421–425 (1966)
Razmyslov, Yu.: Identities of Algebras and Their Representations, Transl. Math. Monogr., vol. 138, Amer. Math. Soc., Providence, RI, 1994
Robinson, D.J.S.: A Course in the Theory of Groups, Springer-Verlag, New York, 1995
Southcott, B.: A basis for the laws of a class of simple groups. J. Austral. Math. Soc. 17, 500–505 (1974)
Strade, H., Farnsteiner, R.: Modular Lie Algebras and Their Representations (Monographs and textbooks in pure and appl. math., vol. 116), Marcel Dekker Inc., New York–Basel, 1988
Thompson, J.: Non-solvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc. 74, 383–437 (1968)
Tokarenko, A.I.: Linear groups over rings. Sibirsk. Mat. Ž. 9, 951–959 (1968) (Russian)
Zorn, M.: Nilpotency of finite groups. Bull. Amer. Math. Soc. 42, 485–486 (1936)
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Bandman, T., Borovoi, M., Grunewald, F. et al. Engel-like characterization of radicals in finite dimensional Lie algebras and finite groups. manuscripta math. 119, 465–481 (2006). https://doi.org/10.1007/s00229-006-0627-0
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DOI: https://doi.org/10.1007/s00229-006-0627-0